Question

let f and g be differentiable functions satisfying f(5)= - 6, f(5)=3, g(5)= - 8, and g(5)=6. find f(5) if f(x)=f(x)/g(x). f(5)=

Explanation:

Step1: Recall quotient - rule

The quotient - rule states that if $F(x)=\frac{f(x)}{g(x)}$, then $F^{\prime}(x)=\frac{f^{\prime}(x)g(x)-f(x)g^{\prime}(x)}{[g(x)]^{2}}$.

Step2: Evaluate $F^{\prime}(5)$

Substitute $x = 5$ into the quotient - rule formula. We know that $f(5)=-6$, $f^{\prime}(5)=3$, $g(5)=-8$, and $g^{\prime}(5)=6$.
$F^{\prime}(5)=\frac{f^{\prime}(5)g(5)-f(5)g^{\prime}(5)}{[g(5)]^{2}}$.
$F^{\prime}(5)=\frac{3\times(-8)-(-6)\times6}{(-8)^{2}}$.

Step3: Calculate the numerator and denominator

First, calculate the numerator: $3\times(-8)-(-6)\times6=-24 + 36=12$.
Then, calculate the denominator: $(-8)^{2}=64$.

Step4: Simplify the fraction

$F^{\prime}(5)=\frac{12}{64}=\frac{3}{16}$.

Answer:

$\frac{3}{16}$